Lecture Notes

The course instructor worked from these notes when presenting each lecture. After the lecture, the notes for the day were posted online for students to access.

1 Elastic Solids; Legendre Transformation; Isotropy; Equilibrium; Compatibility; Constitutive Relations; Variational Calculus; Example of a Functional: String; Extrema - Calculus of Variations; Local Form of Stationarity Condition (PDF)
2 Vainberg Theorem; Hu-Washizu Functional (PDF)
3 Specialized (Simplified) Variational Principles; Hellinger-Reissner Principle; Complementary Energy Principle; Minimum Potential Energy Theorem; Approximation Theory; Rayleigh - Ritz Method (PDF)
4 Weighted - Residuals / Galerkin; Principle of Virtual Work; Geometrical Interpretation of Galerkin’s Method; Galerkin Weighting; Best Approximation Method; The Finite Element Method (PDF)
5 Sobolev Norms; Global Shape Function; Computation of K and fext; Isoparametric Elements (PDF)
6 Higher Order Interpolation; Isoparametric Triangular Elements; Numerical Integration; Gauss Quadrature (PDF)
7 Error Estimation, Convergence of Finite Element Approximations; Error Estimates From Interpolation Theory (PDF)
8 Linear Elasticity; Numerical Integration Errors; Basic Error Estimates; Conditions for Convergence; Patch Test (PDF)
9 Incompressible Elasticity; Hooke’s Law; Governing Equations; “B”-Matrix; Volumetric and Deviatoric Components of “Kh” (PDF)
10 Constraints Ratio; Variational Principle of Incompressible Elasticity; Saddle Point Problem; Constrained Minimization Problem; Reduced Selective Integration; Penalty Formulation (PDF)
11 Assumed Strain Methods; Euler Equations; Mean Dilatation Method; General Expression for Anisotropic Elasticity; Mixed Methods; Discretized Lagrangian (PDF)
12 Finite Elasticity; Metric Changes; State of Stress; Field Equations: Linear Momentum Balance, Angular Momentum Balance, Energy Balance; Nonlinear Elastic Solid (PDF)
13 Variational Formulation; Minimum Potential Energy Principle; Finite Element Approximations; Rayleigh - Ritz Method; Galerkin Approach (PDF)
14 Newton-Raphson Solution Procedure; Continuation Method; Iteration Process; Computation of Tangent Stiffness; Spatial Formulation (PDF)
15 Isoparametric Elements; Commutative Diagram; Tangent Stiffness; Calculation of Tangent Stiffness (continued); Material Frame Indifference; Lagrangian Moduli (PDF)
16 Material Formulation; Specific Material Models; Isotropic Elasticity; Stress-strain Relations; Cayley-Hamilton Theorem; Examples of Constitutive Relations for Finite Elasticity; Saint-Venant / Kirchhoff Model; Mooney-Riulin Model; Neo-Hookean Model Extended to Compressible Range; Computation of Tangent Moduli (PDF)
17 Time Dependent Problems; Nonlinear Elastodynamics (Hyperbolic); Nonlinear Heat Conduction (Parabolic); Initial Boundary Value Problem (IBVP); Finite Element (semi) Discretization (PDF)
18 Constitutive Relations: Fourier Law of Heat Conduction; Finite Element Discretization (Spatial); Time-stepping Algorithms; Newmark Predicators; Newmark Correctors; Convergence Check; Explicit Dynamics (PDF)
19 Trapezoidal Rule - Heat Conduction; Trapezoidal Predictor; Equivalent Static Problem; Trapezoidal Correctors; Convergence Check (PDF)
20 Connection Between Newmark Algorithm and Multistep Methods; Mass Humping; Consistent Mass; Nodal Quadrature; Row (Column) Sum Method; Algorithms Analysis; General Initial Value Problem (IVP) (PDF)
21 Energy Conservation / Dissipation; Abstract Algorithms; Convergence; Conditions of Convergence; Consistency (PDF)
22 Examples: Trapezoidal Rule; Newmark’s Algorithm; Stability; Trapezoidal Rule, Scalar Problem (PDF)
23 Multidimensional Case; Spectral Radius, Lax Equivalence Theorem (PDF)
24 Stability Properties of Trapezoidal Rule; Eigenprojections; Choice of time step; Stability of Newmark’s Algorithm; Iron’s Bounding Principle (PDF - 1.1 MB)
25 Nonlinear Algorithms; Small-strain Plasticity; Kuhn-Tucker Form; Elastic-plastic Moduli; Isotropic-kinematic Hardening (PDF)
26 Time-stepping Algorithms for Constitutive Relations; Numerical Quadrature; Newton-Raphson Solution Procedure; Backward Euler; Geometrical Interpretation; Closest Point Projection Algorithms; J2-isotropic Hardening (PDF)

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